1. Home
  2. Algebra and trigonometry
  3. Equations and inequalities
  4. Other types of equations

2.6 Other types of equations  (Page 4/10)

<< Chapter < Page
  Algebra and trigonometry     Page 4 / 10
Chapter >> Page >

Solve the absolute value equation: | 1 − 4 x | + 8 = 13.

x = −1 , x = 3 2

Got questions? Get instant answers now!

Solving other types of equations

There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.

Solving equations in quadratic form

Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include   x 4 − 5 x 2 + 4 = 0 , x 6 + 7 x 3 − 8 = 0 , and   x 2 3 + 4 x 1 3 + 2 = 0.   In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.

Quadratic form

If the exponent on the middle term is one-half of the exponent on the leading term, we have an equation in quadratic form , which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.

Given an equation quadratic in form, solve it.

  1. Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.
  2. If it is, substitute a variable, such as u , for the variable portion of the middle term.
  3. Rewrite the equation so that it takes on the standard form of a quadratic.
  4. Solve using one of the usual methods for solving a quadratic.
  5. Replace the substitution variable with the original term.
  6. Solve the remaining equation.

Solving a fourth-degree equation in quadratic form

Solve this fourth-degree equation:   3 x 4 − 2 x 2 − 1 = 0.

This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let   u = x 2 . Rewrite the equation in u .

3 u 2 − 2 u − 1 = 0

Now solve the quadratic.

3 u 2 − 2 u − 1 = 0 ( 3 u + 1 ) ( u − 1 ) = 0

Solve each factor and replace the original term for u.

3 u + 1 = 0 3 u = −1 u = − 1 3 x 2 = − 1 3 x = ± i 1 3
u − 1 = 0 u = 1 x 2 = 1 x = ±1

The solutions are   ± i 1 3   and   ± 1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve using substitution:   x 4 − 8 x 2 − 9 = 0.

x = −3 , 3 , − i , i

Got questions? Get instant answers now!

Solving an equation in quadratic form containing a binomial

Solve the equation in quadratic form:   ( x + 2 ) 2 + 11 ( x + 2 ) − 12 = 0.

This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting   u = x + 2.   Then rewrite the equation in u.

u 2 + 11 u − 12 = 0 ( u + 12 ) ( u − 1 ) = 0

Solve using the zero-factor property and then replace u with the original expression.

u + 12 = 0 u = −12 x + 2 = −12 x = −14

The second factor results in

u − 1 = 0 u = 1 x + 2 = 1 x = −1

We have two solutions:   −14 , and −1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve:   ( x − 5 ) 2 − 4 ( x − 5 ) − 21 = 0.

x = 2 , x = 12

Got questions? Get instant answers now!

Solving rational equations resulting in a quadratic

Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.

Questions & Answers

calculate molarity of NaOH solution when 25.0ml of NaOH titrated with 27.2ml of 0.2m H2SO4
Gasin Reply
what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
Valentina Reply
definition of the periodic table
Cosmos Reply
What is the lkenes
Da Reply
what were atoms composed of?
Moses Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask